The thing with the Hawks is that they were always a bunch of incoherent underachievers with an average to bad coach. I remember everyone saying: "If Josh Smith could get brain surgery, he'd be a superstar!" or "If Iso Joe ever learned the new word in the dictionary (pass), he'd be a fantastic piece!" or "If Horford didn't have glass bones, he'd be a huge factor!" or even "If Teague puts it together he'd be pretty good!"
The difference now is the consensus that the team is overachieving due to great coaching and players that fit together. However, they don't fit the typical overachieving-underdog narrative because they weren't terrible to start; they were mediocre. So instead of being an expected lottery team that is fighting for the 8 seed (like the Celtics), they were an expected 6-8 seed that has the 2nd best record in the league.
Now, the argument here is whether or not the Celtics can maintain this scrappy, fluid style of play with the many roster changes to come. Teams like the Spurs have been doing it for seemingly forever, but they had a pretty darn good start with Timmy Duncan and The Admiral. The Hawks are a story because, as the OP said, they had great roster upheaval and achieved success relatively quickly.
From what I can gather, the main disagreement stems from the fact that most rebuilds seem to have two steps: allocation of talent and putting it all together. For both, you have to get lucky. Teams like the Thunder had amazingly good fortune in amassing talent, but they haven't had a coach or the health to "put it together." Same goes for other teams like the Rockets. The Spurs have both, which is why they're champions. The old Heat had so much talent that there were so many ways to "get it together" so quickly. The 2008 Celtics are an anomaly because they got a huge influx of talent and "put it together" almost immediately. Some people are arguing that we are "putting it together" now, and we will naturally get assets through good management and free agency. Others argue that we don't have enough talent and talent allocation has to come first, so we should tank.
I, for one, believe the Celtics can do it (meaning the former) because of their foundation. It took the Hawks years to amass talent (via tanking), and it took them even longer to find a coach to put it all together. I think with Stevens already here and Danny as the GM, the Celtics will find ways to slowly (or quickly) get better while not taking steps back like other teams. Not to be discounted is the ownership group's resources; the Celtics won't lose a talent like Harden to avoid the luxury tax.